Hexagonal Numbers

Whole numbers can be categorized by the shapes that arrays of objects of a number make. For example, every non-prime number can be arranged as a rectangular array. Similarly, some numbers, such as 3, 6, and 10 can be formed into triangular arrays. In the game of ten-pins (bowling), the ten pins are aranged into a triangular shape.

Of particular interest are the hexagonal numbers, those that can be arranged into a hexagonal array as shown below:

The first four hexagonal numbers.

Hexagonal arrays are useful for arranging elements into a close-packed array in two dimensions, such as in the placement of feed horns in some communication antenna systems. Of particular interest are the large percentage of prime numbers that the hexagonal numbers form. The table below gives some of the geometric and other properties of the first 20 integers.

Number  Triangle  Square  Hexagon  Cube  Hex Pyram  Mersene   H Factors  M Factors
1       1         1       1        1     1          3         prime      prime
2       3         4       7        8     8          5         prime      prime
3       6         9       19       27    27         9         prime      3, 3
4       10        16      37       64    64         17        prime      prime
5       15        25      61       125   125        33        prime      3, 11
6       21        36      91       216   216        65        7, 13      5, 13
7       28        49      127      343   343        129       prime      3, 43
8       36        64      169      512   512        257       13, 13     prime
9       45        81      217      729   729        513       7, 31      3, 3, 3, 19
10      55        100     271      1000  1000       1025      prime      5, 5. 41
11      66        121     331      1331  1331       2049      prime      3, 683
12      78        144     397      1728  1728       4097      prime      17, 241
13      91        169     469      2197  2197       8193      7, 67      3, 2731
14      105       196     547      2744  2744       16385     prime      5, 29, 113
15      120       225     631      3375  3375       32769     prime      3, 3, 11, 331
16      136       256     721      4096  4096       65537     7, 103     prime
17      153       289     817      4913  4913       131073    19, 43     3, 43691
18      171       324     919      5832  5832       262145    prime      5, 13, 37, 109
19      190       361     1027     6859  6859       524289    13, 79     3, 174763
20      210       400     1141     8000  8000       1048577   7, 163     17, 61681

The formula for the triangular numbers is

T(n) = (n² + n) / 2

The formula for the square numbers is, of course, just n². The formula for the hexagonal numbers utilizes the formula for the triangular numbers:

H(n) = 6 * T(n - 1) + 1

A hexagonal pyramid is built by stacking incrementally smaller hexagons. It turns out that the volume of a hexagonal pyramid of size n is exactly equal to the cube of n:

P(n) = n³

The factors of the first 20 hexagonal numbers are shown in the column labeled "H Factors." As can be seen, there are more prime hexagonal numbers than rectangular ones. The trend continues through the first fifty hexagonals as shown in this Excel file.

For comparison, the first 20 Mersene numbers (of the form M(n) = 2ⁿ + 1) are shown. Their factors are shown in the column labeled "M Factors." As can be seen, the hexagonal primes appear to be more numerous than the Mersene primes.

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